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Unformatted text preview: STAT 333 Random Walk On A Circle The ordinary random walk (which we have studied in great detail) can be modified in various ways by the introduction of different kinds of boundaries. The introduction of boundaries typically changes the long-run behaviour of the process in some significant way. For example, the gamblers ruin can be viewed as a random walk in which we have placed absorbing boundaries at 0 and N . The process eventually hits one of the boundaries and stops (unlike the ordinary walk which keeps bouncing around forever). A different type of example occurs when we place a reflecting boundary at 0 in the random walk and alter the state space to be S = { , 1 , 2 , 3 ,... } . Here, whenever the process hits 0, the process bounces (or reflects) back to state 1 at the next step (otherwise it behaves like an ordinary random walk). You might want to think about the long-run behaviour of this process for different values of p . The Circular Random Walk Another way of introducing a boundary into the ordinary random walk is to fix an integer m 2 and make the identification m 0, then carry out arithmetic mod m . This is equivalent to taking nodes 0 , 1 , . .., m- 1 and arranging them in a circle. The process then jumps one node clockwise1 and arranging them in a circle....
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